Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

TL;DR

This paper investigates the regularizing effects and decay properties over time of solutions to a class of parabolic equations with superlinear first-order gradient terms, considering unbounded initial data.

Contribution

It provides new decay estimates and regularity results for parabolic problems with superlinear gradient terms and unbounded initial data, extending existing theories.

Findings

Establishes regularizing effects for solutions.

Derives decay rates for short and long times.

Handles unbounded initial data in specific Lebesgue spaces.

Abstract

We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partial\Omega,\\ u(0,x)=u_0(x) &\text{in}\,\, \Omega. \end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive, bounded and measurable matrix, the growth rate $q$ of the gradient term is superlinear but still subnatural, $\gamma>0$, the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^\sigma(\Omega)$ for $\sigma=\sigma(q,p,N)$.